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6CCVD Research

Benchmarking machine learning algorithms for adaptive quantum phase estimation with noisy intermediate-scale quantum sensors

At a Glance

Section titled “At a Glance”
MetadataDetails
Publication Date2021-06-03
JournalEPJ Quantum Technology
AuthorsNelson Filipe Costa, Yasser Omar, Aidar Sultanov, Gheorghe Sorin Paraoanu, Nelson Filipe Costa
InstitutionsArt Research Centre of the Slovak Academy of Sciences, Instituto Politécnico de Lisboa
Citations17
AnalysisFull AI Review Included

Executive Summary

Section titled “Executive Summary”
  • Core Value Proposition: Classical machine learning (ML) algorithms (Differential Evolution, DE; Particle Swarm Optimization, PSO) are successfully benchmarked to optimize adaptive quantum phase estimation (QPE) using non-entangled, noisy intermediate-scale quantum (NISQ) sensors.
  • Performance Achievement: The adaptive ML protocols drive measurement precision close to the Standard Quantum Limit (SQL), achieving Holevo Variance (VH) scaling approximated by VH ~ N-α, with αDE = 0.75 and αPSO = 0.64 in the ideal case (N < 15).
  • Noise Robustness: ML-derived policies demonstrate superior robustness compared to non-adaptive protocols, particularly above a critical noise threshold (e.g., high Gaussian noise or low visibility v = 0.6).
  • Optimization Target: The algorithms optimize the sequential phase feedback policy (xm) to minimize the Holevo Variance (VH), a direct measure of estimation precision for periodically bounded variables.
  • Applicability: The protocols are directly relevant for experimental platforms including Mach-Zehnder optical interferometers, superconducting qubits (transmons), trapped ions, and nitrogen-vacancy (NV) centers in diamond.
  • Computational Constraint: Simulations were limited to N < 25 qubits/measurements due to the polynomial scaling of complexity (O(N3)) and imposed iteration limits (G = 100), which prevented full convergence at larger N.

Technical Specifications

Section titled “Technical Specifications”

The following data points summarize the performance metrics and physical parameters used in the benchmarking simulations and discussed experimental implementations.

ParameterValueUnitContext
Simulation Qubit Count (N)5 to 25QubitsRange tested for benchmarking algorithms
Ideal SQL Scaling (αSQL)1N-αStandard Quantum Limit VH ~ N-1
DE Performance Scaling (αDE)0.75N-αIdeal noiseless scenario (N < 15)
PSO Performance Scaling (αPSO)0.64N-αIdeal noiseless scenario (N < 15)
Optimal DE Amplification (F)0.7DimensionlessParameter for mutation stage
Optimal DE Crossover (C)0.8DimensionlessParameter for diversity stage
Optimal PSO Learning Rates (α, β)0.8, 0.8DimensionlessDesirability to follow personal/global best
Optimal PSO Max Velocity (Vmax)0.2DimensionlessControls step size and convergence
Decoherence Visibility (v)0.6DimensionlessVisibility corresponding to optimal sensing time (τ = T2/2)
Superconducting Qubit T210µsTypical decoherence time
Superconducting Qubit Sensing Time (τ)1µsUsed in simulation, resulting in v = 0.9 (Gaussian noise)
NV Center Zero-Field Splitting (D)2.87GHzEnergy separation between ms = 0 and ms = ±1 states

Key Methodologies

Section titled “Key Methodologies”

The adaptive QPE scheme relies on sequential Ramsey interferometry where classical ML algorithms determine the optimal feedback policy.

  1. Adaptive Quantum Phase Estimation (QPE) Scheme:

    • Qubits are sequentially injected (in state |0> or |1>) into a circuit consisting of two Hadamard gates (H) sandwiching a controllable phase shift gate (Uφ,θ).
    • The phase shift Uφ,θ is defined by the unknown phase (φ) and the controllable phase (θm).
    • The measurement outcome (ζm) is used to update the controllable phase for the next qubit.

  2. Policy Optimization Target:

    • The performance of a policy (the vector of phase instructions x = {x1, x2, …, xN}) is evaluated by minimizing the Holevo Variance (VH).
    • VH is calculated numerically by averaging the sharpness S = |(ei(φ-θ))| over K training instances (K = 10N2).

  3. Adaptive Update Rule (Markovian Feedback):

    • The controllable phase θm is updated based on the previous measurement outcome (ζm-1) and the policy instruction xm: θm = θm-1 ± xm.
    • The ML algorithms search for the optimal set of N phase instructions {x1, …, xN} that minimizes VH.

  4. Machine Learning Algorithms:

    • Differential Evolution (DE): Uses mutation and crossover stages to generate new candidate policy vectors, selecting the vector that minimizes VH for the next generation. Convergence is guaranteed when the amplification parameter F is strictly less than the crossover parameter C (F < C).
    • Particle Swarm Optimization (PSO): Particles (candidate solutions) move in the N-dimensional search space, adjusting their velocity based on their personal best position (pbest) and the collective global best position (gbest). Convergence is guaranteed when the global learning rate β is greater than or equal to the personal learning rate α (β ≥ α).

  5. Noise Benchmarking Scenarios:

    • Gaussian Noise (GSN): Applied to the controllable phase shifter (θm) following a normal distribution with standard deviation σ.
    • Random Telegraph Noise (RTN): Applied to θm, causing a discrete offset (λ) with a fixed probability (η = 0.4).
    • Quantum Decoherence: Modeled by reduced interference visibility (v = exp[-τ/T2]), affecting the measurement probabilities P±(ζm|φ, θm).

Commercial Applications

Section titled “Commercial Applications”

This research provides robust optimization strategies for high-precision measurements in several quantum technology sectors, particularly those relying on NISQ devices.

  • Quantum Sensing and Metrology:

    • High-Precision Magnetometry: Optimization of Ramsey interference protocols using solid-state qubits (NV centers in diamond) and superconducting circuits (transmons) for ultra-sensitive magnetic field detection.
    • Robust Sensing: ML policies mitigate noise effects (Gaussian, RTN, decoherence) inherent in real-world sensors, enabling reliable operation in non-ideal environments.

  • Optical Interferometry:

    • Phase Detection: Implementation in Mach-Zehnder interferometers for highly accurate phase-shift estimation, relevant for applications like gravitational wave detectors and advanced optical navigation systems.

  • NISQ Quantum Computing:

    • Algorithm Subroutines: The optimized QPE scheme is a fundamental component of larger quantum algorithms, such as the inverse Quantum Fourier Transform (QFT) used in Shor’s factorization.
    • Molecular Spectroscopy: Potential application in computing molecular spectra, a phase-estimation problem.

  • Solid-State Qubit Technology (Relevant to CVD Diamond):

    • NV Center Devices: The robustness against RTN and decoherence is critical for NV centers, which are used for nanoscale sensing (e.g., magnetic imaging of biological samples or integrated circuits) where long coherence times are essential.

Tech Support

Section titled “Tech Support”

Original Source

Section titled “Original Source”

References

Section titled “References”
  1. 2011 - Probabilistic and statistical aspects of quantum theory [Crossref]
  2. 2016 - Mathematical methods of statistics (PMS-9)
  3. 1995 - Quantum measurement

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